I've been thinking about this for around 7 months now, which is way too much; Munkres seems like the "typical" introduction to topology book (kind of like how Griffiths is the "typical" E&M text), and various people (from the reviews over at Amazon) make it seem like it is an undergraduate level text. In the textbook however, it says otherwise:

It also claims that the only (main) prerequisite is set theory and maybe some analysis to fill in the gaps in the first part.

Is Munkres a good introductory topology text? And yes, I do have experience with set theory.

Munkres is usually the standard undergrad text for Math majors. Alternatives you may consider include 'Introduction to Topology' by Bert Mendelson, very good introduction and Dover publications make it quite a bit cheaper than Munkres.

The other alternative that comes to mind is Introduction to Topology and Modern Analysis by Simmons.

Munkres is not a grad level text and is actually quite easy. However, for some reasons, I do not like Munkres, mainly because it ommits nets and filters (aside from some silly exercises in some chapter), because (as a consequence) it makes the Tychonoff theorem way harder than it could be, because he only gives a limited number of counterexamples (counterexamples of course are not topology, but they can be immensely beautiful) and because the exercises tend to be too easy to my taste (then again, I would probably not have this objection if I first started topology).

On the other hand, Munkres covers a wealth of material in point-set topology which I rarely see in many other books, including many metrization theorems and generalization of the Ascoli-Arzela theorem.

To be honest, there are very little topology books that I really like and most are grad level. For example, Willard and Kelley are my favorites, but they are hardly suitable for an undergrad. Munkres, Simmons, Lee (intro to topological manifolds) are all ok.
If you are feeling up for a challenge, then I do highly recommend http://www.pdmi.ras.ru/~olegviro/topoman/ This is a problem-based approach (if you buy the book, then the solutions can be found at the end of the book). So if you're not in a rush to learn topology, then there is probably not a better book than this one.

Another extremely good book is Brown https://www.amazon.com/Topology-Groupoids-Ronald-Brown/dp/1419627228 The first few chapters deal with point-set topology and it has many good exercises. However, it soon starts dealing with a very nonstandard approach (but probably superior approach) to algebraic topology.

Some analysis books also deal with topology. For this I mention Knapp's Basic Analysis.

Another non-textbook, but actually my favorite topology book is Steen and Seebach: https://www.amazon.com/Counterexamples-Topology-Dover-Books-Mathematics/dp/048668735X/ The book contains tons of very beautiful and elegant counterexamples. So if you ever wondered for an example of a nonseparable compact Hausdorff space, it's in the book. Definitely a book worth looking through.

I've decided to get Munkres and Steen and Seebach. S&S seems like another language to me at the moment but I'll hopefully understand it in a bit. If I'm not satisfied w/ Munkres then Brown it is. Also, I'm curious as to what area of math you specialize in, Micromass. You seem to know a little bit of everything.

I don't like Munkres very much either, especially not at its $150 price on Amazon!

For point set topology, there is a far superior sequence using only Dover titles (hence dirt cheap):

Mendelson, Introduction to Topology - very nice intro, doesn't cover all that much but gets you efficiently to the basics on compactness and connectedness. Only costs $7!

Gemignani, Elementary Topology - covers similar ground to Munkres plus some topics he omits, such as nets and filters. Very readable in my opinion. Also only $7!

Willard, General Topology - if you're feeling very ambitious This one is a whopping $20 but the exercises may keep you busy (or haunt you) for a lifetime.

I'm sorry for being so late to this thread, but I was wondering if you could tell me what you think would be the best book for my background. I have taken a course (really a crash course) in Topology based off the book by Munkres. At first, I didn't understand it, and I even thought the basic/introductory set theory problems were hard. Since then however, I have gained significant mathematical maturity, and I think I could go back through that book with absolutely no problem at all. I still know the some of the main definitions and concepts behind point set topology like what a topology is, bases, common types of topologies, metric spaces, separation axioms, Urysohn's lemma etc. I want to relearn topology, but I am wondering if it is even worth it to read through Munkres' book at this point. I would like to know what book covers the most material (and of course in at least an adequate manner) and is hopefully at the graduate level. One I have heard about is by Dugundji. How does that compare to Kelley and Willard? I also have taken interest in that problem-based book and like the idea of learning by doing stuff myself so I really might consider using it, but does it cover as much material as other books like Dugundji? Can it replace other books?